Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,602 questions
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Modal models as reduced products?
In model theory for standard first-order logic, one constructs a single model, a reduced product, from a collection of first-order models, together with an index set and a filter on the index set.
In ...
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Intersection of cocompact closed normal subgroups
Let $G$ be a locally compact Hausdorff topological group.
Definition A closed normal subgroup $H \unlhd G$ is called cocompact if $G/H$ is compact with respect to the quotient topology.
Note that ...
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Direct proof of "K is projective iff C(K) has the Hahn-Banach property" ?
An object $X$ of a given category is called projective if for each morphism $f : X \rightarrow Z$, and each epimorphism $ g : Y \twoheadrightarrow Z$, there is a morphism $h : X \rightarrow Y$ such ...
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Nonhomeomorphic CW-complexes that are "stably" homeomorphic
Do there exist CW-complexes $X$ and $Y$ that are not homeomorphic, but $X \times I$ and $Y \times I$ are homeomorphic? Here $I$ denotes the unit interval $[0, 1]$.
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Questions about spectra of rings of continuous functions
I have been thinking a bit about rings of continuous functions of various kinds -- how they motivate the more modern notion of the Zariski topology on the prime spectrum as well as how they fit into a ...
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measurable function on a locally compact space for a regular measure
A well known classical fact is that a Lebesgue measurable function on Euclidean space is almost everywhere equal to a Baire class 2 function. A relatively modern reference for this fact is van Rooij -...
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Test functions with small support and nonnegative Fourier transform
The following problem arose in a question I recently asked : given a (possibly non abelian) compact group $G$ and a neighbourhood $U$ of the identity in $G$, can we always find a function $f : G \...
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How much $\beta \mathbb{N}$ is homogenous?
Let $p,q\in \beta \mathbb{N}\setminus \mathbb{N}$. Must always the spaces $\beta \mathbb{N}\setminus \{p\}$ and $\beta \mathbb{N}\setminus \{q\}$ be homeomorphic? If no, can we for each point $p\in \...
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Characterization of cocompact group action
Wikipedia claims the following:
In mathematics, an action of a group G on a topological space X is cocompact if the quotient space X/G is a compact space or, equivalently, if there is a compact ...
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Which is the correct ring of functions for a topological space?
There is a fact that I should have learned a long time ago, but never did; I was reminded that I did not know the answer by Qiaochu's excellent series of posts, the most recent of which is this one.
...
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The identity element of a compact group is a limit point of any "polynomial sequence"
Is there an "elementary" (say ultrafilter-free) proof of the following fact: if $G$ is a compact (Hausdorff) topological group, if $g \in G$ is any element from this group, and if $P$ is a polynomial ...
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the space of maximal ideals in C(X) and C*(X) [closed]
Let $C(X)$ be the continous function ring and $C*(X)$ be the bounded continous function ring.$Max C(X)$ consisting of all maximal ideals in $C(X)$.
Question:why $Max C(X)$ and $Max C*(X)$ are compact ...
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Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group? [xpost from math.SE]
And what else can be said, if so?
(Original math.SE post)
In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. (...
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Geometry Realization of Homology Class
Hello!
My question is about the realization of homology class.
The definition of the realizaion of homology class is: for manifold M and a homology class $z\in H_k(M)$, k is an integer. If we find a ...
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functional subrings
I should recall the notion of maximal subring of a commutative unitary ring $R$.
Def: A commutative ring $S$ is called a maximal subring of $R$ if $S \subset R$ and if $T \subset R$ constitute a ...
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a questions about the sums of intersections of maximal ideals
why the z-ideals in C(X) are basically the sums of intersections of maximal ideals?
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C*-algebras with no nontrivial endomorphisms
Pick a C*-algebra $A$ and call a (*-)endomorphism $\alpha:A\to A$ nontrivial if it is injective and $\alpha(A)\neq A$.
Question: Do there exist infinite dimensional C*-algebras with no nontrivial ...
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H-spaces without rational homology
Does there exist a simply connected, non-contractible manifold $M$, which is an $H$-space,
and whose rational homology groups vanish in positive degrees?
My space $M$ is in fact homotopy equivalent ...
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A question about Q?
Let A=$\{a_n : n\in \omega \}\subset 2^{\omega\times\omega}$ be nonempty countable without isolated points (i.e. homeomorphic to $\mathbb{Q}$), and satisfy $ \forall n\in \omega \exists^\infty m|\{k:...
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Arbitrary union of meagre open sets
Let $X$ be a topological space. A subset $M$ of $X$ is called meagre (or of first category) if it is covered by the union of a countable family of closed subsets of $X$ with empty interior.
Can you ...
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When is a sublevel set path-connected?
I am trying to completely characterize the conditions on $f : \mathbb{R}^n \to \mathbb{R}$ under which $\{x | f(x) \le 0 \}$ is path-connected. There are many obvious conditions that are sufficient (...
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Closure properties of familes of $G_\delta$ sets.
Given a family of sets $G\subset P(X)$, can one characterize by "closure properties" alone whether or not $G$ arises as the family of all $G_\delta$ for some topology on $X$? some Polish space ...
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When is $\{ x \ge 0 | f(x) \le 0\}$ path-connected?
I'm trying to determine the conditions on $f : \mathbb{R}^n_{\ge 0} \to \mathbb{R^n}$ under which $\{ x \ge 0 | f(x) \le 0 \}$ is path-connected. We can assume that $f$ is continuous and concave.
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A closed connected component in a topological space does not contain any path-connected subset?
Does there exist such a non-trivial closed connected component U of some connected topological space X or a non-trivial connected topological space X that do not contain any non-trivial path-connected ...
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Arbitrary small positive lower semi continuous functions
This question is a generalization of the question posed in this page to lower semi continuous functions. so let me describe the Question in the following way.
Def: Let $(X,\tau)$ be a Tychonoff ...
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A question on minimal idempotent ultrafilter on N^2
Is there some minimal idempotent ultrafilter $q \in \beta( \mathbb{N}^2)$ (with respect to the law $"+"$) such that any $A \in q$ is a subset of $\mathbb{N} \times \{ 0 \} $ ?
(See for example http:/...
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De Rham representatives of the cohomology classes in $H^*(SU(3))$
I would like to know differential forms representing the cohomology classes of $SU(3)$. I know that there exist a unique bi-invariant form in each class, but I'm not highly motivated by simply putting ...
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product spaces of rationals
Let $Q$ follow subspace topology from $R$
Then I think it is true that $Q^n$ and $Q^m$ (with product topology) are not homeomorphic.I also think it will be possible to define "rational" homotopy ...
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Thurston-Bennequin number vs. checkerboard coloring difference
For an alternating knot K, checkerboard-color the knot (if this is a lousy ASCII crossing: %, white goes to the left/right and black to top/bottom). Assume no surplus Reidemeister 1
kinks exist (K has ...
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Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
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$\beta\mathbb{N}$ vs $\beta\mathbb{Z}$
Just started learning the Stone-Cech compactification of discrete groups this week. My motivation comes from a question on $\beta\mathbb{Z}$. Surprisingly, I realized there are muchhhh more literature ...
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Cantor Sets Inside Cantor Sets
(Or: "I heard you liked Cantor Sets...")
I'm working on a student project, and the following construction came up very naturally: If $C$ is the usual Cantor Set, build a countable union of copies of ...
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A basic question on Stone-Cech compactification of $\mathbb{Z}$
Can the identity isomorphism on the additive group $\mathbb{Z}$ be extended to a non-identity semigroup isomorphism on $\beta\mathbb{Z}$, and still preserves $\beta\mathbb{Z}\setminus\mathbb{Z}$? ...
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Characterization of Stone-Cech compactifications
Suppose I have an infinite discrete topological space $X$ of cardinality $\kappa$. Then I know some things about the Stone-Cech compactification, $\beta X$: it is Hausdorff and compact but not ...
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Sums of Strongly z-ideals
In the rings of continuous functions,i.e.$(C(X))$ an ideal $I$ is called strongly $z$-ideal if it is an intersection of some maximal ideals of $C(X)$. i.e. $$I=\cap_{\alpha \in A} \mathcal{M_{\alpha}}$...
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Can Reidemeister 3 be weakened?
If you take the diagram of the Reidemeister 3 move and "shortcircuit" two ends, you get (click https://i.sstatic.net/gfOKy.jpg if Imgur hotlink doesn't work):
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Borel sets preserved under open maps?
Given open map f: $R^n$ to $R^n$ such that each open set $U\in R^n$, $f(U)$ is also open. Are Borel sets in $R^n$ preserved under f?
Motivation: Pre-image of Borel sets under continuous map is a ...
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Non trivial definition of bicontinuous functions and the ring of all bicontinuous functions.
At first let me recall that if There are two topology $\tau_1$and $\tau_2$ on a set $X$, the triple $(X,\tau_1,\tau_2)$ is called a bitopological space.
There are many definitions and properties ...
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A space in which sequences have unique limits but compact sets need not be closed
A topological space is KC if every compact subspace is closed.
A topological space is US if every convergent sequences has exactly one limit.
Does someone know an easy example of a US space which is ...
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Topologies on the field of rationals
Ostrowski's theorem give the answer for valuations, but is there a complete classification of (at least separated) topologies on Q (compatible with the field operations, obviously)?
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The set of Upper semi-continuous functions as a ring.
I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set
{$[a , b): a,b \in \mathbb{R} $} as it's base.
If $X$ is a topological space, an upper ...
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When is Prim(A) of an infinite discrete group hausdorff ?
Does anyone know, if the following result has been proved ?
Let G be an infinite discrete group. A = L1(G) it's algebra and Prim(A) the set of prime ideals with spectral topology.
The result is :
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Function spaces over pseudocompact spaces
Let $K$ be a pseudocompact Tychonoff space that is, a Hausdorff $T_{3.5}$ space for which every continuous function $\varphi \colon K\to \mathbb{C}$ is bounded. Let $\beta$ be the Stone-Cech extension ...
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Lifting identities of formal power series
I am looking for a possibly general class of algebraic structures (maybe special topological rings) in which one can deduce identities of concrete power series from formal ones. This class should ...
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Assumptions on a commutative C*-algebra to get a nice C(X) - space
I have the following question,
Is it possible to get somehow a compact Hausdorff space $X$ which is second-countable from a unital commutative C*-algebra. If it is possible, what should we assume ...
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Homeomorphism of the rationals
In working with the classification of stable vector bundles on $\mathbb{P}^2$, I've found that I need to answer a fairly basic question from analysis/point set topology. Here it is.
Suppose $f:\...
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Technique: Compactness => (Finite -> Reals)
Context
I'm studying a classical results of Erdos and Lovasz, on colorings of the real line.
The theorem to be proved is as follows:
Let $m, k$ be two positive integers satisfying:
$$e(m(m-1)+1)k\...
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Staggered timing on 2-D random walks by multiple agents
In 2-D lattice random walks by multiple drunks who can't step onto each other, mathematically I would just say the whole cellular automaton updates "at once".
But to simulate this on a computer, I ...
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The difference between a sequential space and a space with countable tightness
Hi, I have recently encountered these two definitions of a sequential space and a space of countable tightness.
And I seem to have difficulty understanding what is the difference between these two ...
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Rugged manifold
It is well known that any compact smooth $m$-manifold can be obtained from $m$-ball by gluing some points on the boundary.
Is it still true for topological manifold?
Comments:
To proof the smooth ...
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